Differentiability at the edge of the percolation cone and related results in first-passage percolation

Abstract

We study first-passage percolation in two dimensions, using measures μ on passage times with b: = inf supp(μ) > 0 and \({\mu(\{b\})=p\geq \vec p_c}\) , the threshold for oriented percolation. We first show that for each such μ, the boundary of the limit shape for μ is differentiable at the endpoints of flat edges in the so-called percolation cone. We then conclude that the limit shape must be non-polygonal for all of these measures. Furthermore, the associated Richardson-type growth model admits infinite coexistence and if μ is not purely atomic the graph of infection has infinitely many ends. We go on to show that lower bounds for fluctuations of the passage time given by Newman–Piza extend to these measures. We establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for μ. This result confirms a prediction of Newman and Piza (Ann Probab 23:977–1005, 1995) and Zhang (Ann Probab 36:331–362, 2008). Under the assumption of finite radius of curvature for the limit shape in these directions, we obtain a power-law lower bound for the variance and an inequality between the exponents χ and ξ.